Gauge and the Hole Argument

In this paper I shall extend the formulation of the hole argument to permutable theories. As covariant theories provides a general mathematical framework for classical physics, permutable theories provide the language for quantum physics. This analogy is deeply founded: as covariant theories are defined functorially on the category of manifolds and local diffeomorphisms with values into the category of fibered manifolds and fiber-preserving morphisms, and some rule of selecting sections([5]), permutable theories are functorially defined on the category of sets and permutations with values into the category of fibered sets and fiber-preserving automorphisms, and rules of selecting sections. One of the main features of covariant theories, in particular general relativity, is that the field equation possesses gauge freedom associated with global diffeomorphisms of the underlying manifold. I shall explain here how the hole argument is a reflection of this gauge freedom. Finally I shall point out some implications of the hole argument to individuation problem of basic objects in physics.